Abstract

Nonlinear principal component analysis (NLPCA) was used for compression and reconstruction of the total radiance factors (TRFs) of fluorescent samples. The spectral dataset included a total of 358 fluorescent reflectance spectra in the visible range of the spectrum. Spectral data compression was followed by extracting the parameterized nonlinear manifolds using the NLPCA technique. To compare the performance of NLPCA-based compression with the linear method, the orthonormal feature vectors of the dataset were also extracted by the linear PCA. The spectral performance of NLPCA and PCA-based compression approaches was assessed by the root mean square error and the goodness-fitting coefficient between the real and the reconstructed spectra. The percentages of feasible spectra by each method, i.e., those with nonnegative TRFs, were also reported as other criteria for the evaluation of methods. Furthermore, the colorimetric performance of methods were appraised by the measuring the CIELAB 1976 color difference values between the actual and reconstructed spectra under illuminants D65 and A and the 1964 standard observer. The NLPCA-based compression method performed significantly better than the standard PCA-based technique particularly in the lower dimensional spaces of the spectral radiance factors of fluorescent colors.

© 2013 Optical Society of America

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References

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  1. D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
    [CrossRef]
  2. J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  3. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. 3, 29–33 (1986).
    [CrossRef]
  4. H. S. Fairman and M. H. Brill, “The principal components of reflectance,” Color Res. Appl. 29, 104–110 (2004).
    [CrossRef]
  5. F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvectors,” J. Opt. Soc. Am. A 23, 2020–2026 (2006).
    [CrossRef]
  6. K. Ansari, S. H. Amirshahi, and S. Moradian, “Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique,” Color Technol. 122, 128–134 (2006).
    [CrossRef]
  7. F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
    [CrossRef]
  8. T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
    [CrossRef]
  9. S. H. Amirshahi and F. Agahian, “Basis functions of the total radiance factor of fluorescent whitening agents,” Text. Res. J. 76, 192–207 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (1)

R. Jafari, S. H. Amirshahi, and S. A. Hosseini Ravandi, “Spectral analysis of blacks,” Color Res. Appl. 37, 176–185 (2012).
[CrossRef]

2011 (1)

2008 (2)

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

2006 (3)

S. H. Amirshahi and F. Agahian, “Basis functions of the total radiance factor of fluorescent whitening agents,” Text. Res. J. 76, 192–207 (2006).
[CrossRef]

K. Ansari, S. H. Amirshahi, and S. Moradian, “Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique,” Color Technol. 122, 128–134 (2006).
[CrossRef]

F. Ayala, J. F. Echavarri, and P. Renet, “Use of three tristimulus values from surface reflectance spectra to calculate the principal components to reconstruct these spectra by using only three eigenvectors,” J. Opt. Soc. Am. A 23, 2020–2026 (2006).
[CrossRef]

2005 (1)

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

2004 (2)

H. S. Fairman and M. H. Brill, “The principal components of reflectance,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

G. Kerschen and J. C. Golinval, “Feature extraction using autoassociative neural networks,” Smart Mater. Struct. 13, 211–219 (2004).

2002 (1)

G. Ketschen and J. C. Golival, “Non-linear generalization of principal component analysis: from a global to a local approach,” J. Sound Vib. 254, 867–876 (2002).
[CrossRef]

1999 (1)

F. Del Frate and G. Schiavon, “Non-linear principal component analysis for the radiometric inversion of atmospheric profile by using neural networks,” IEEE Trans. Geosci. Remote Sens. 37, 2335–2342 (1999).
[CrossRef]

1997 (1)

1992 (1)

1991 (1)

M. A. Kramer, “Non-linear principal component analysis using autoassociative neural networks,” AIChE J. 37, 233–243 (1991).
[CrossRef]

1989 (1)

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst. 2, 303–314 (1989).
[CrossRef]

1986 (1)

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. 3, 29–33 (1986).
[CrossRef]

1964 (1)

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).

Agahian, F.

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

S. H. Amirshahi and F. Agahian, “Basis functions of the total radiance factor of fluorescent whitening agents,” Text. Res. J. 76, 192–207 (2006).
[CrossRef]

A. Rayat, S. H. Amirshahi, and F. Agahian, “Compression of spectral data using Box-Cox transformation,” Color Res. Appl.; doi: 10.1002/col.21771, to be published. (First published online October 11, 2012.)

Amirshahi, S. A.

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Amirshahi, S. H.

R. Jafari, S. H. Amirshahi, and S. A. Hosseini Ravandi, “Spectral analysis of blacks,” Color Res. Appl. 37, 176–185 (2012).
[CrossRef]

S. Peyvandi and S. H. Amirshahi, “Generalized spectral decomposition: a theory and practice to spectral reconstruction,” J. Opt. Soc. Am. A 28, 1545–1553 (2011).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

S. H. Amirshahi and F. Agahian, “Basis functions of the total radiance factor of fluorescent whitening agents,” Text. Res. J. 76, 192–207 (2006).
[CrossRef]

K. Ansari, S. H. Amirshahi, and S. Moradian, “Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique,” Color Technol. 122, 128–134 (2006).
[CrossRef]

A. Rayat, S. H. Amirshahi, and F. Agahian, “Compression of spectral data using Box-Cox transformation,” Color Res. Appl.; doi: 10.1002/col.21771, to be published. (First published online October 11, 2012.)

S. Farajikhah, F. Madanchi, and S. H. Amirshahi, “Non-linear principal component analysis for compression of spectral data,” presented at IS2011, Ljubljana, Slovenia, 2011.

Ansari, K.

K. Ansari, S. H. Amirshahi, and S. Moradian, “Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique,” Color Technol. 122, 128–134 (2006).
[CrossRef]

Ayala, F.

Berns, R. S.

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Brill, M. H.

H. S. Fairman and M. H. Brill, “The principal components of reflectance,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

Cohen, J. B.

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).

Cybenko, G.

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst. 2, 303–314 (1989).
[CrossRef]

Del Frate, F.

F. Del Frate and G. Schiavon, “Non-linear principal component analysis for the radiometric inversion of atmospheric profile by using neural networks,” IEEE Trans. Geosci. Remote Sens. 37, 2335–2342 (1999).
[CrossRef]

Echavarri, J. F.

Fairman, H. S.

H. S. Fairman and M. H. Brill, “The principal components of reflectance,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

Farajikhah, S.

S. Farajikhah, F. Madanchi, and S. H. Amirshahi, “Non-linear principal component analysis for compression of spectral data,” presented at IS2011, Ljubljana, Slovenia, 2011.

Garcia-Beltran, A.

Golinval, J. C.

G. Kerschen and J. C. Golinval, “Feature extraction using autoassociative neural networks,” Smart Mater. Struct. 13, 211–219 (2004).

Golival, J. C.

G. Ketschen and J. C. Golival, “Non-linear generalization of principal component analysis: from a global to a local approach,” J. Sound Vib. 254, 867–876 (2002).
[CrossRef]

Harifi, T.

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Hernandez-Andres, J.

Hosseini Ravandi, S. A.

R. Jafari, S. H. Amirshahi, and S. A. Hosseini Ravandi, “Spectral analysis of blacks,” Color Res. Appl. 37, 176–185 (2012).
[CrossRef]

Jafari, R.

R. Jafari, S. H. Amirshahi, and S. A. Hosseini Ravandi, “Spectral analysis of blacks,” Color Res. Appl. 37, 176–185 (2012).
[CrossRef]

Kerschen, G.

G. Kerschen and J. C. Golinval, “Feature extraction using autoassociative neural networks,” Smart Mater. Struct. 13, 211–219 (2004).

Ketschen, G.

G. Ketschen and J. C. Golival, “Non-linear generalization of principal component analysis: from a global to a local approach,” J. Sound Vib. 254, 867–876 (2002).
[CrossRef]

Kramer, M. A.

M. A. Kramer, “Non-linear principal component analysis using autoassociative neural networks,” AIChE J. 37, 233–243 (1991).
[CrossRef]

Kruger, U.

U. Kruger, J. Zhang, and L. Xie, “Developments and applications of non-linear principal component analysis—a review” [Online]. Available: http://pca.narod.ru/1MainGorbanKeglWunschZin.pdf .

Madanchi, F.

S. Farajikhah, F. Madanchi, and S. H. Amirshahi, “Non-linear principal component analysis for compression of spectral data,” presented at IS2011, Ljubljana, Slovenia, 2011.

Maloney, L. T.

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. 3, 29–33 (1986).
[CrossRef]

Moradian, S.

K. Ansari, S. H. Amirshahi, and S. Moradian, “Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique,” Color Technol. 122, 128–134 (2006).
[CrossRef]

Nakano, M.

Nakauchi, S.

Peyvandi, S.

Rayat, A.

A. Rayat, S. H. Amirshahi, and F. Agahian, “Compression of spectral data using Box-Cox transformation,” Color Res. Appl.; doi: 10.1002/col.21771, to be published. (First published online October 11, 2012.)

Renet, P.

Romero, J.

Schiavon, G.

F. Del Frate and G. Schiavon, “Non-linear principal component analysis for the radiometric inversion of atmospheric profile by using neural networks,” IEEE Trans. Geosci. Remote Sens. 37, 2335–2342 (1999).
[CrossRef]

Tzeng, D. Y.

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Usui, S.

Xie, L.

U. Kruger, J. Zhang, and L. Xie, “Developments and applications of non-linear principal component analysis—a review” [Online]. Available: http://pca.narod.ru/1MainGorbanKeglWunschZin.pdf .

Zhang, J.

U. Kruger, J. Zhang, and L. Xie, “Developments and applications of non-linear principal component analysis—a review” [Online]. Available: http://pca.narod.ru/1MainGorbanKeglWunschZin.pdf .

AIChE J. (1)

M. A. Kramer, “Non-linear principal component analysis using autoassociative neural networks,” AIChE J. 37, 233–243 (1991).
[CrossRef]

Color Res. Appl. (4)

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

H. S. Fairman and M. H. Brill, “The principal components of reflectance,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

R. Jafari, S. H. Amirshahi, and S. A. Hosseini Ravandi, “Spectral analysis of blacks,” Color Res. Appl. 37, 176–185 (2012).
[CrossRef]

Color Technol. (1)

K. Ansari, S. H. Amirshahi, and S. Moradian, “Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique,” Color Technol. 122, 128–134 (2006).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

F. Del Frate and G. Schiavon, “Non-linear principal component analysis for the radiometric inversion of atmospheric profile by using neural networks,” IEEE Trans. Geosci. Remote Sens. 37, 2335–2342 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. 3, 29–33 (1986).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Sound Vib. (1)

G. Ketschen and J. C. Golival, “Non-linear generalization of principal component analysis: from a global to a local approach,” J. Sound Vib. 254, 867–876 (2002).
[CrossRef]

Math. Control Signals Syst. (1)

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst. 2, 303–314 (1989).
[CrossRef]

Opt. Rev. (1)

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Psychon. Sci. (1)

J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).

Smart Mater. Struct. (1)

G. Kerschen and J. C. Golinval, “Feature extraction using autoassociative neural networks,” Smart Mater. Struct. 13, 211–219 (2004).

Text. Res. J. (1)

S. H. Amirshahi and F. Agahian, “Basis functions of the total radiance factor of fluorescent whitening agents,” Text. Res. J. 76, 192–207 (2006).
[CrossRef]

Other (5)

A. Rayat, S. H. Amirshahi, and F. Agahian, “Compression of spectral data using Box-Cox transformation,” Color Res. Appl.; doi: 10.1002/col.21771, to be published. (First published online October 11, 2012.)

U. Kruger, J. Zhang, and L. Xie, “Developments and applications of non-linear principal component analysis—a review” [Online]. Available: http://pca.narod.ru/1MainGorbanKeglWunschZin.pdf .

S. Farajikhah, F. Madanchi, and S. H. Amirshahi, “Non-linear principal component analysis for compression of spectral data,” presented at IS2011, Ljubljana, Slovenia, 2011.

http://www.nlpca.org/matlab.html,19/5/2013 .

MATLAB, Version 7.8.0, The MathWorks Inc. (2009).

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Figures (8)

Fig. 1.
Fig. 1.

Autoassociative network architecture for implementation of NLPCA. σ shows sigmoidal neurons and * indicates sigmoidal or linear nodes [11].

Fig. 2.
Fig. 2.

TRFs of 358 fluorescent samples.

Fig. 3.
Fig. 3.

Numbers of samples recovered in different GFC categories by the PCA and NLPCA methods. (a) Unacceptable, (b) acceptable, (c) good, and (d) excellent reconstruction.

Fig. 4.
Fig. 4.

Percentage of recovered feasible spectra by the PCA and NLPCA methods.

Fig. 5.
Fig. 5.

Results of spectral recovery of six randomly selected fluorescent samples by six basis functions using classical as well as NLPCA techniques.

Fig. 6.
Fig. 6.

Results of spectral recovery of six randomly selected fluorescent samples by nine basis functions using standard as well as NLPCA techniques.

Fig. 7.
Fig. 7.

Results of spectral recovery of six randomly selected fluorescent samples by 12 basis functions using classical as well as NLPCA techniques.

Fig. 8.
Fig. 8.

Differences between the actual and reconstructed spectra of fluorescent samples obtained with (a) standard PCA and (b) NLPCA methods by using three basis functions.

Tables (3)

Tables Icon

Table 1. Mean, Maximum, and Standard Deviations of RMS Errors between the Actual and Reconstructed Spectra with PCA and NLPCA Methods Using Different Dimensional Subspaces

Tables Icon

Table 2. Mean and Minimum of GFCs of Recovery by the PCA and NLPCA Methods, Using Different Dimensional Subspaces

Tables Icon

Table 3. Mean, Maximum, and Standard Deviation of Color Difference Values between the Actual and the Reconstructed Spectra with the PCA and NLPCA Methods under A and D65 Illuminants and 1964 Standard Observera

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

Σ=1n1i=1n(rir¯)(rir¯)T,
Λ=V1ΣV,
ΣV=VΛ,
r[v1,v2,,vq]c+r¯,
PC=G·R,
R^=H·PC.
vk=j=1N2wjk2σ(i=1N1wij1ui+θj1).
σ(x)=11+ex

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